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DISCONTINUITY SETS of SOME INTERVAL BIJECTIONS E. De International Journal of Pure and Applied Mathematics Volume 79 No. 1 2012, 47-55 AP ISSN: 1311-8080 (printed version) url: http://www.ijpam.eu ijpam.eu DISCONTINUITY SETS OF SOME INTERVAL BIJECTIONS E. de Amo1 §, M. D´ıaz Carrillo2, J. Fern´andez S´anchez3 1,3Departamento de Algebra´ y An´alisis Matem´atico Universidad de Almer´ıa 04120, Almer´ıa, SPAIN 2Departamento de An´alisis Matem´atico Universidad de Granada 18071, Granada, SPAIN Abstract: In this paper we prove that the discontinuity set of any bijective function from [a, b[ to [c, d] is infinite. We then proceed to give a gallery of examples which shows that this is the best we can expect, and that it is possible that any intermediate case exists. We conclude with an example of a (nowhere continuous) bijective function which exhibits a dense graph in the rectangle [a, b[ × [c, d] . AMS Subject Classification: 26A06, 26A09 Key Words: discontinuity set, fixed point 1. Introduction During our investigations [1] concerning on the application of Ergodic Theory of Numbers to generate singular functions (i.e., their derivatives vanish almost everywhere), we came across a bijection from a closed interval to another that it is not closed: c : [0, 1] −→ [0, 1[, given by (segments on some intervals): (n − 1)! n! − 1 1 c (x) := x − + n n2 n + 1 1 1 1 1 which maps 1 − n! , 1 − (n+1)! onto n+1 , n and c(1) = 0. h h h h c 2012 Academic Publications, Ltd. Received: April 19, 2012 url: www.acadpubl.eu §Correspondence author 48 E. de Amo, M.D. Carrillo, J.F. S´anchez Such a function provides very interesting properties as: i. c preserves arithmetical properties for x; n n n ii. for composition c := c ◦ ···◦ c and for all x, the sequence (c (n))n has an associated density function (a.d.f.) which is a singular function; and iii. this function c allows us to show an example of a non measurable Lebesgue set. After this finding, we asked ourselves if there were other examples of bijec- tions f of this explicit nature given by a formula and if their properties had been studied or not. The literature we have read up to this moment is richer from the theoretical than the exemplary point of view (see [2] and [3]): a. such a function f will never be continuous on all its domain of definition; b. explicit examples of such bijections f are not exhibited After this stage, questions concerning the nature of the set D(f) of points where f is not continuous arise in a natural way. Some interesting facts that are well known with respect to this discussion (see [2]) are the following: (α) If f is monotonic, then D(f) is a denumerable set (and, conversely, for each denumerable set A there exists a monotonic function f such that A = D(f)). (β) There is no f such that R\Q = D(f) (it is a consequence of the Baire Category Theorem). (γ) There exists f for which D(f) is the Cantor Ternary Set. (δ) For any f, D(f) is a denumerable union of closed sets (D(f) ∈ Fσ for short), and conversely, for all E ∈ Fσ, there is some function f such that E = D(f). (Moreover, if the set E is closed, then the function f can be taken bounded.) (ǫ) The set D(f) is of first category (i.e. it is a denumerable union of sets M ◦ such that M is empty) if and only if f is continuous at a dense set. DISCONTINUITY SETS OF SOME INTERVAL BIJECTIONS 49 A simple and basic fact grows up among all these ideas, if f is a bijection: clearly D(f) 6= ∅; but, what more can be said concerning its size? We give a simple but, at the same time, complete solution in the sense that we prove the following: i. For each bijection f : [a, b[ −→ [c, d] , the set D(f) is infinite. ii. There are bijections f, g : [a, b[ −→ [c, d] such that on the one hand D(f) is denumerable and on the other hand D(g) = [a, b[ . As a colophon, we introduce an example (the 8th) which flirts with the classical, but always up-to-date, idea of space-filling curves (see [4]): we carry out (of course, very far from the situation D(f) = ∅, now f will be nowhere continuous!) a bijection from [0, 1[ to [0, 1], such that its graph is dense in the unit square [0, 1]2. Moreover we propose an Open Question to investigate whether a bijection f : [a, b[ −→ [c, d] exits or not, such that the set D(f) is denumerable and has derivatives at a finite number of points. 2. The Result Elementary topological techniques in Real Analysis show that one-to-one and onto continuous mappings between intervals on the real line R are homeomor- phisms; i.e. f and f −1 are continuous bijections. (Such a function is forced to be strictly monotonic.) A first consequence of this fact is that if one of the intervals is open then the other one is open too. Thus, we have the following and well-known fact: Fact. There are no continuous bijections from [a, b[ to [c, d]; i.e. these intervals are not homeomorphic. Let us denote by D(f) the set of points where f is not continuous (or discontinuous), i.e. D(f) is the discontinuity set of f. We are interested in its size. We will consider, without loss of generality, bijections from [0, 1[ to [0, 1]. Let N denote the natural numbers, i. e. the set Z+ of all positive integers. Firstly, we prove that for such bijections the set D(f) is infinite. Theorem. No continuous bijection exists from [0, 1[ to [0, 1] but a finite number of points, i.e. {f : [0, 1[ −→ [0, 1] | f bijection and D(f) finite} = ∅. 50 E. de Amo, M.D. Carrillo, J.F. S´anchez Proof. By Reductio ad absurdum, we suppose there exists a bijection f : [0, 1[ −→ [0, 1] , that is continuous but at points in a finite set. Let us denote the set {x0,x1, ..., xp} := D(f) ∪ {0}. We have no loss of generality if we consider the lexicographic order (n<m ⇒ xn < xm) on this finite set. (Note that x0 = 0.) Let xp+1 := 1, and denote Ik := ]xk−1,xk[, for k = 1, 2, ..., p + 1. Hence, there is a collection {f (Ik) : k = 1, 2, ..., p + 1} of (nonempty and) pairwise disjoint open intervals, i.e. [0, 1] \ {f (Ik) : k = 1, 2, ..., p + 1} = {y0,y1, ..., yp,yp+1} . But this yields a contradiction: it is impossible that {f (x0) ,f (x1) , ..., f (xp)} and {y0,y1, ..., yp,yp+1} to be equipotent sets. 3. A Gallery of Examples Among infinite sets, denumerable ones are the smallest. The next example presents a particular f that will be the smoothest we can expect to encounter. Example 1. There exists a bijection f from [0, 1[ to [0, 1] having (a con- stant) derivative at each point but a denumerable set in [0, 1[. Let f : [0, 1[ −→ [0, 1] be given by 1, x = 0 1 N f(x) := x, x ∈ ]0, 1[ \ n+1 : n ∈ 1 n o 1/n, x ∈ n+1 : n ∈ N n o This function is one-to-one and onto. Moreover, it is continuous on ]0, 1[ \ 1 n+1 : n ∈ N , because the local character of continuity. The same goes for n o ′ 1 derivability. In fact, f (x) = 1 for all x ∈ ]0, 1[ \ n+1 : n ∈ N . Furthermore, 1 n o f is discontinuous on each x ∈ n+1 : n ∈ N ∪ {0}. The way to define the functionn f in Exampleo 1 suggests a suitable method to find an explicit bijection from [0, 1] on [0, 1] ∩ (R \ Q) . The rest of the paper is devoted to exhibiting more examples showing that many intermediate cases are possible. DISCONTINUITY SETS OF SOME INTERVAL BIJECTIONS 51 In the above example f ′ ([0, 1[ \ D(f)) splits to one point. This idea is very far from being the paradigm. Example 2. There exists a bijection f from [0, 1[ to [0, 1] such that for all r> 0 there exist x ∈ [0, 1[ such that f ′ (x)= r. Let f : [0, 1[ −→ [0, 1] be given by 0, x = 0 f(x) := n n−1 n−1 1 n−1 n N ( [n (n + 1)] x − n + n+1 , x ∈ n , n+1 ,n ∈ i i Easy calculations show that, for each natural n: n − 1 n 1 1 f , = , n n + 1 n + 1 n n−1 n (in particular, one-to-one) and each one of the families n , n+1 : n ∈ N 1 1 ni i o and n+1 , n : n ∈ N is a collection of mutually disjoint intervals and ni i o +∞ n − 1 n +∞ 1 1 ]0, 1[ = ∪ , and ]0, 1] = ∪ , ; n=1 n n + 1 n=1 n + 1 n thus, f is one-to-one and onto from [0, 1[ to [0, 1]. n−1 n Moreover, this function has derivative at each point x ∈ n , n+1 ,n ∈ N, and satisfies i h n − 1 n f ′ , = ]0,n[ . n n + 1 We omit small modifications in f, for the sake of simplicity we omit them, which would allow us to obtain a bijection such that {f ′(x) : f has derivative at x} = R.
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